Washington is 8-11-0 this season in games when it ends the period on a downer and 4-4-1 when it gives up a goal in the frame's first minute. You would expect those types of goals to have a damaging effect on the hearts and minds of the team that gives them up. After all, momentum appears to instantly swing in that team's favor. But do teams gain any real advantage when scoring a goal in the first or last minute of a period?
Not really. The win percentage for a team scoring one of these “momentum-shifting” goals (empty nets excluded) is not significantly different than a coin flip, except for when it is scored in the last minute of the second frame. Teams that score in the middle period’s waning moments see a meaningful bump in their win percentage and those that score during the last 60 seconds of the game's final period in regulation win fewer games when compared to other periods.
Even if the "momentum-shifting" goal is the first goal of the game, we don't see much evidence of an actual momentum shift.
Want to shift momentum in your team's favor? Just score; it really doesn't matter when.